Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(7,8)(9,10,12,14)(11,13,15,16)(18,22)(23,25)(24,29)(30,31);(1,2,6,4)(3,7,5,8)(18,22)(20,28)(23,25)(26,32);(1,3,6,5)(2,7,4,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(1,4,6,2)(3,7,5,8)(9,11,12,15)(10,13,14,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,5,6,3)(2,7,4,8)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41490, #7640201, #7640228, … |
| maximal normal subgroup of | C2×#41490, #7640201, #7640228, … |
| maximal quotient of | #544715, #545207, #545701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34478);